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Mathematical Association of America Golden Section Mathematical Association of America Golden Section (Northern California, Nevada and Hawaii) Saturday, February 27, 2016 University of California, Davis STUDENT POSTER SESSION ABSTRACTS Title. Dehn{Sommerville Relations and the Catalan Matroid Authors: Anastasia Chavez, University of California, Berkeley; Nicole Yamzon, San Francisco State University Faculty sponsor. Dr. Federico Ardila, San Francisco State University Abstract. The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P d+1 is simplicial the Dehn{Sommerville relations condense the f-vector into the g-vector, which has length d 2 e. Thus, to determine the f-vector of P , we only need to know approximately half of its entries. This raises the d+1 question: Which (d 2 e)-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid. Title. Algebraic Structures of 2-Dimensional Topological Quantum Field Theories Author. Joseph Dominic, Saint Mary's College of California Faculty sponsor. Dr. Weiwei Pan, Saint Mary's College of California Abstract. In this poster we study the mathematical aspects of 2-dimensional topological quantum field theories (2D TQFT's) culminating with a classification theorem which characterizes 2D TQFT's with a certain class of algebraic structures called Frobenius Algebras. We then attempt to construct a generalized 2-dimensional topological quantum field theory which classifies Hopf Algebras and examine the difficulties inherent to this endeavor. Finally, we study a relationship between Hopf Algebras and Frobenius Algebras. Title. Jacobi-Type Sums With an Explicit Evaluation Modulo Prime Powers for Small Powers of Primes Author. Franciska Domokos, California State University, Sacramento Faculty sponsor. Dr. Vincent Pigno, California State University, Sacramento Abstract. Given two mod q Dirichlet Characters χ1 and χ2 the classical Jacobi Sum is q X J(χ1; χ2; q) := χ1(x)χ2(1 − x): x=1 W. Zhang and W. Yao found that the classical Jacobi sum has explicit evaluations as long as q is a perfect square and Zhang & Xu obtained an evaluation of multivariate Jacobi sum for certain squareful q (if p j q, then p2 j q). This result was extended by Alsulmi, Pigno & Pinner to the general Jacobi-Type sum pm pm X X ··· χ1(x1) ··· χs(xs); (1) x1=1 xs=1 k1 ks m A1x1 +···+Asxs ≡B mod p for general squareful q using the sum reduction methods of Cochrane and Zheng as long as m ≥ 2t + n + 2. Here B is an integer, p - A1A2 and the integers t and n are defined by n ti p jj B and t = maxftig where p jj ki; i = 1; : : : ; s: i In this project we evaluated the sum (1) on a smaller range t + n + 1 < m ≤ 2t + n + 2 when s = 2 and p is an odd prime. Title. Enumerating Lattice 3-Polytopes Author. M´onicaG´omez,University of California, Davis Faculty sponsor. Dr. Jes´usDe Loera, University of California, Davis 3 Abstract. A lattice 3-polytope is a polytope P ⊂ R with integer vertices. We call size of P the number of lattice points it contains, and width of P the minimum, over all integer linear functionals f, of the length of the interval f(P ). In this poster I present my results on a full enumeration of lattice 3-polytopes via their size and width: for any fixed n ≥ d + 1 there are only finitely many lattice 3-polytopes of width larger than one and size n, and this finite list can be partially obtained from the list of size n − 1. The rest of the polytopes can be separately classified due to some special features. Title. A Sampling Kaczmarz-Motzkin Algorithm for Linear Feasibility Author. Jamie Haddock, University of California, Davis Faculty sponsor. Dr. Jes´usDe Loera, University of California, Davis Abstract. We combine two algorithmic techniques for solving systems of linear inequalities, the relaxation method of Agmon, Motzkin et al. and the randomized Kaczmarz method. In doing so, we obtain a family of algorithms that generalize and extend both techniques. While we prove similar convergence results, our computational experiments show our algorithms often vastly outperform the original methods. Title. Numerical Ranges of Composition Operators Whose Symbols are Disk Automorphisms Author. Buddy Galleti, Christopher Hurley*, and Adam Mair, Cal Poly, San Luis Obispo Faculty sponsor. Dr. Jonathan Shapiro, Cal Poly, San Luis Obispo Abstract. We examine properties of the numerical ranges of certain composition operators acting on the Hardy space. We pay particular attention to those composition operators whose symbols are rotations and elliptic automorphisms of the disk. Title. The Missing Piece: The Effect of DNA Curvature on the Topological Structure of Kinetoplast DNA Minicircle Networks Author. Lara Ibrahim, University of California, Davis Faculty sponsor. Dr. Javier Arsuaga, University of California, Davis Abstract. The kinetoplast DNA (kDNA) of Trypansoma Brucei forms a chain-mail like network of thousands of catenated minicircles. Little is known about the biophysical factors which maintain this complex network. Experimental data suggests that each minicircle is singly linked to 3 other minicircles (i.e. valence 3). However, mathematical modeling which factors in minicircle orientation, volume exclusion, and electrostatic repulsions fails to reduce the valence below 11. To account for this discrepancy between the experimental and theoretical valence number, we are analyzing the sequence dependent curvature of the minicircle sequence. We suggest that intrinsic bending or \kinking" of the DNA strand may allow for such low valence. We employ the use of the program 3DNA, which builds visualizations of the three dimensional structure of nucleic acids based on specified base pair parameters. Title. Algebraic Methods for Neural Codes Author. Nida Kazi, San Jose State University Faculty sponsor. Dr. Elizabeth Gross, San Jose State University Abstract. A neural code is a collection of codewords (0-1 vectors) of a given length n; it captures the co-firing d patterns of a set of neurons. A neural code is convexly realizable if there exist n convex sets in some R so that each codeword in the code corresponds to a unique intersection carved out by the convex sets. There are some methods to determine whether a neural code is convexly realizable, however, these methods do not describe how to draw a realization. In this work, we construct toric ideals from neural codes, and we show how we can use these ideals, along with the theory of inductive piercings and Euler diagrams, to draw realizations for particular classes of codes. Title. The Length Spectrum of the Sub-Riemannian Three-Sphere Author. David Klapheck, California State University, Sacramento Faculty sponsor. Dr. Michael VanValkenburg, California State University, Sacramento Abstract. We determine the lengths of all closed sub-Riemannian geodesics of the three-sphere S3. Our methods do not require us to directly construct solutions of the geodesic equations. Title. A Rainbow Ramsey Analogue of Rado's Theorem Author. R. N. La Haye, A. Montejano, D. Oliveros, and E. Roldn-Pensado, University of California, Davis Faculty sponsor. Dr. Jes´usDe Loera, University of California, Davis Abstract. We present a Rainbow Ramsey version of the well-known Ramsey-type theorem of Richard Rado. We use new techniques from the Geometry of Numbers. We also disprove two conjectures proposed in the literature. Title. Linear Systems and Chip Firing on Graphs Author. Bo Lin, University of California, Berkeley Faculty sponsor. Dr. Bernd Sturmfels, University of California, Berkeley Abstract. The linear system jDj of a divisor D on a metric graph has the structure of a cell complex. We introduce the anchor divisors and anchor cells in it - they serve as the landmarks for us to compute the f-vector of the complex and find all cells in the complex. A linear system can also be identified as a tropical convex hull of rational functions. We compute the extremal generators of the tropical convex hull using the landmarks. We apply these methods to some examples - namely the canonical linear systems on K4 and K3;3. Title. The Monstrous Poster Author. Alexander Lowen, Saint Mary's College of California Faculty sponsor. Dr. Kristen Beck, Saint Mary's College of California Abstract. J. H. Conway constructs the monster group using the Leech Lattice. The purpose of my research is to exposit on the Hexacode and the extended Golay Code in order to construct the Leech Lattice. In doing so, I hope to instill an intuition for Conway's second sporadic group, Co1, which is needed to construct the monster group. Title. Generalized Eulerian Numbers and the δ-polynomial for Half-Open Lattice Parallelepipeds: A Geometric Perspective Authors: Emily McCullough*, San Francisco State University; Dr. Katharina Jochemko, Vienna University of Technology Faculty sponsor. Dr. Matthias Beck, San Francisco State University Abstract. The coefficients of the δ-polynomial of a lattice polytope P encode information about the lattice point counts in positive integer dilates of P. We consider the relationship between the (B; `)-Eulerian num- bers, a refined descent statistic on the set of signed permutations on f1; 2; : : : ; dg, and the δ-polynomial for the d-dimensional half-open ±1-cube with ` non-translate facets removed. Using the interplay of geometry and com- binatorics, we improve upon known inequality relations on the coefficients of the δ-polynomials for specific fam- ilies of half-open parallelepipeds. Our results extend naturally to the δ-polynomials for closed lattice zonotopes. Title. Transportation Polytopes and their Hierarchy of Circuit Diameters Author. Jacob Miller, University of California, Davis Faculty sponsor. Dr. Jes´usDe Loera, University of California, Davis Abstract. The study of the diameter of the graph of polyhedra is a classical problem in the theory of linear programming.
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